What is the ruling on future profits from a permissible project, after investing money obtained from forbidden binary options trading, and donating the original capital and half of the profits, knowing that the trading began with a gift of $1?

It is not permissible to deal in binary options. If one acquires unlawful wealth, they must dispose of it by giving it to the poor or spending it on public welfare, unless they are in need, in which case they may take from it what suffices their need, or what can serve as capital for a permissible trade, or take it as a loan and then repay it. If you dispose of the amount of the unlawful capital, you have fulfilled your obligation, and any profits that come to you thereafter are permissible for you.